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I always get punctuation in quotes wrong, so I pulled out my trusty Chicago Manual of Style (15th edition. It's so solid. So orange. So satisfying to beat people with.) and asked it.
It said that periods and commas always go inside the quotes, and other punctuation always goes outside the quotes, unless the other punctuation is part of the quote. It suggested that if this is a problem, you can move over everything to the Oxford style, where only punctuation that is part of the quote goes inside the quotes, but that this requires "extreme authorial precision" and works best with single quotation marks.
I'm gonna stand by "periods and commas inside the quotes." If you'd like to disagree, I've got this big orange book I can beat you with...